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Master’s Thesis
Portfolio optimization with KOSPI200 and House
price index
Minky Cho
Department of Mathematical Sciences
Graduate School of UNIST
Portfolio optimization with KOSPI200 and House
price index
Minky Cho
Department of Mathematical Sciences
Abstract
In this paper, we will investigate optimal investment and consumption strategies in the market with CRRA utility function. In this market, the investor chooses a portfolio that consists of one bond (with zero interest rates), one illiquid risky asset (having transaction costs), and one liquid risky asset (having no transaction costs). Using shadow price which is the virtual price between the bid and ask price, we can derive the optimal investment and consumption strategies.
From these results we obtained, we will apply to the market having stock and residential real estate. For the analyze, we chose KOSPI200 (the index consists of 200 big companies of the Stock Market Division in Korea) for the stock and house price index (the price of the residential real estate in Korea) for the real estate. With KOSPI200 and house price index data, we obtained the optimal investment and consumption strategies and also checked the effect of the transaction costs on two risky assets and consumption.
Contents
I Introduction . . . 1
II Model . . . 4
2.1 The market . . . 4
2.2 Utility function . . . 5
2.3 Shadow price approach . . . 6
III Review of Choi [1] . . . 9
3.1 Frictionless market with two risky assets . . . 9
3.2 Result of Choi [1] . . . 12
IV Asymptotic ally optimal trading and consumption strategies . . . 15
4.1 Optimal strategy . . . 16 V Parameter Estimation . . . 18 5.1 Data description . . . 18 5.2 Results . . . 20 VI Conclusion . . . 28 References . . . 28
I
Introduction
In one of his seminal papers, Merton solved an optimal investment and consumption problem in the market where the investor allocate his wealth between a liquid asset(stock) and a risk-free asset to maximize his utility function. This problem is called Merton’s portfolio problem. The objective of this problem is maximizing the following expectation.
E Z T 0 e−δtu(ct) +γe−δTu(WT) where T ∈[0,∞]
Here,δ is discount rate,ctis consumption rate at time t,Wtis his wealth at time t,is desired level of bequest. And utility function u is of the constant relative risk aversion (CRRA) form:
u(c) = c p p, c≥0 and U(0) = 0, p > 0, −∞ p ≤0.
In this frictionless market where there are one risk-free asset and one risky asset with CRRA utility function, Merton showed that the optimal allocation strategy is investing a constant proportion of wealth in the risky asset. The line whose slope is a proportion derived from Merton is called Merton’s line. When the value process is finite, Merton’s portfolio problem with an infinite horizon is reduced to maximizing the following expectation.
E
Z ∞
0
e−δtu(ct)dt
Figure 1 shows the optimal strategy in the frictionless market and in the market with trans-action costs. The x-axis denotes the sum of the risky and risk-free asset value and the y-axis denotes risky asset. So, the slope means the proportion of risky asset. In the market with trans-action costs, the optimal strategy is defined by two straight lines. Merton’s line lies between these two straight lines. Inside the two straight lines, the optimal strategy for the investor is not trading. Outside of the no transaction region, the optimal strategy for the investor is adjusting his portfolio to the nearest line so that the proportion could be in the no transaction region.
Many kinds of researches have been done by many researchers to remove the frictionless market condition. That is, many researchers have tried to consider transaction costs. Also, there were some trials considering not one risky asset but two risky assets. So in this paper, we consider the model with one risky asset with transaction costs and one risky asset having no transaction costs.
[1], [2] solved this problem with one bond and two risky assets. One is the liquid asset that has no transaction costs and the other is an illiquid asset that has proportional transaction costs. To solve this problem, [1] used the shadow price which is the virtual price between the bid and ask prices of the original market. This shadow price approach makes the original optimization problem solved in the frictionless market. So using the shadow price approach, he derived the optimal investment and consumption strategies by solving the optimization problem in the frictionless market.
[1] and [2] heuristically showed that the HJB equation can be transformed into a free boundary problem with a first order differential equation. The main difference between [1] and [2] is that [2] just analyzed the HJB equation but [1] took the dual approach. But since they are actually solving the same problem, we only focus on the [1].
In this paper, using the optimal investment and consumption strategies obtained in [1], we applied the results to the real data. We considered two risky assets, stock and residential real estate assets which represents liquid and illiquid asset respectively. the stock also has the transaction cost but many papers consider it as a liquid asset because the transaction cost of it is small. To analyze the market with stock and real estate, we used two monthly data associated with them. One is the KOSPI200 (index consists of 200 big companies of the Stock Market Division in Korea) and the other is the house price index(the price of the residential real estate in Korea) from January 2004 to August 2019.
To apply the result, we have to estimate the parameters µ1, µ2, σ1, σ2, ρ, δ, p, λ. Using the
KOSPI200 and house price index and other related data, we can estimateµ1, µ2, σ1, σ2, ρ, λ. The
way how we estimated will be discussed in section 5. But parameters δ, λ is hard to estimate using these data. So we followed [3], [4] to estimate δ, λrespectively.
For estimating the parameter δ, we followed the method used in [3]. In [3], they estimated the discount factorβ asβ= 1/(1 + ˜r) where˜r is real interest rate which is the sum of nominal interest rate and expected inflation rate. β andδ have the following relation:
e−δ =β = 1/(1 + ˜r) So using this, we can estimate theδ.
For estimating the parameterp, we just used the value estimated in [4]. In [4], they estimated the risk aversion coefficients of real estate asset owners based on the theory of consumption based capital asset pricing model(CCAPM). For the CRRA utility function, estimated relative risk aversion coefficients were 3.85 in 2012. In other words, since (1−p) = 3.85, we used the valuep=−2.85.
The remainder of the paper is organized as follows. In Section 2, we will discuss the model and formulate the problem. And In section 3, we will derive the optimal investment and consumption strategies in the frictionless market with two risky assets and briefly review the paper [1]. In the next section, Asymptotic ally optimal trading and consumption strategies obtained from [1] will be introduced without the proof. And finally, the optimal strategies using KOSPI200 and house price index data will be shown.
II
Model
2.1 The market
We will consider the market with one bond having zero interest rates and two risky assets. A frequently used model for modeling risky asset prices is the geometric Brownian motion. IfS(i)
denotes the price process of each risky asset, then S(i) follows a geometric Brownian motion if it satisfies the following stochastic differential equation.
dS(i)=S(i) µidt+σidB (i) t , S0(i)>0, i= 1,2 (1)
Here, µi and σi are positive constants and B(1) and B(2) are standard Brownian motion with correlationρ∈(−1,1). The information structure is given by the augmented filtration generated byB(1) andB(2). We assume thatS(1) is a risky asset having the proportional transacion costs (i.e. illiquid asset) andS(2)is a risky asset having no transacion costs (i.e. liquid asset). In other words, S(1) has proportional transaction costs whenever the investor trades S(1). Specifically, for the illiquid asset price process St(1), there exists λ >¯ 0, λ ∈ (0,1) such that buying and selling price can be represented asS¯t(1) = (1 + ¯λ)St(1) and St(1) = (1−λ)St(1).
Let η0, η1, η2 be the investor’s initial share of bond, illiquid asset and liquid asset. And
similarly, let the triple (ϕ(0)t , ϕ(1)t , ϕ(2)t ) the number of shares in the bond, illiquid asset and liquid asset at time t. Also, let ct be the consumption rate at time t. To include the possible case of the initial jump, we set (ϕ(0)0−, ϕ(1)0−, ϕ(2)0−) = (η0, η1, η2) and right-continuous after that.
First, we define the two conditions, self-financing and admissibility.
Definition 2.1.1. A strategy (ϕ(0), ϕ(1), ϕ(2), c) is said to be self-financing if
ϕ(0)t +ϕ(2)t St(2) =η0+η2S0(2)+ Z t 0 ϕ(2)u dSu(2)− Z t 0 ¯ Su(1)d(ϕ(1)u )↑+ Z t 0 S(1)u d(ϕ(1)u )↓− Z t 0 cudu (2)
where(ϕ(1)u )↑ and(ϕ(1)u )↓ are the cumulative numbers of illiquid asset bought and sold up to time t.
Definition 2.1.2. A self-financing strategy (ϕ(0), ϕ(1), ϕ(2), c) is called admissible if
ϕ(0)t +S(1)t (ϕ1t)+−S¯t(1)(ϕ(1)t )−+St(2)ϕ(2)t ≥0, t≥0 (3) We assume that
η0+S(1)t (η1)+−S¯t(1)(η1)−+St(2)η2≥0.
for the initial admissiability.
Assume that ϕ(1) is of finite variation a.s. The set of admissible strategies is denoted by A
and the set of all c such that (ϕ(0), ϕ(1), ϕ(2), c)∈ Afor some (ϕ(0), ϕ(1), ϕ(2))is denoted by C.
2.2 Utility function
For p ∈(−∞,1), we consider the constant relative risk aversion (CRRA) type utility function
U : [0,∞]→[−∞,∞) U(c) = c p p, c≥0 and U(0) = 0, p > 0, −∞ p≤0.
Our goal is to analysis the optimal consumption and investment problem
sup c∈CE Z ∞ 0 e−δtU(ct)dt , (4)
where the constantδ denotes discount factor. Because people prefer the present than the future, the discounting term is multiplied to the utility function.
To make the optimization problem (4) well-posed, we assume the two conditions below.
(1) δ > qµ2 2σ22 and δ > q(2µ1(1 +q)−σ21) 2(1 +q)2 (2) µ1 6= ρµ2σ1 σ2 andµ2 6= ρσ1σ2 1 +q.
2.3 Shadow price approach
In this subsection, we introduce the shadow price which is used in [1]. Simply speaking, the shadow price is the virtual price which is between the bid price and ask price satisfying its maximal expected utility is equal to the maximal utility of the original problem. The shadow price approach makes the transaction cost problem to be solved by constructing a frictionless market model. To define the shadow price process, we will first define the set of the consistent price process.
Definition 2.3.1. The set of consistent price processes S is defined as
S=
˜
S: ˜S is an Ito-process, and S(1)t ≤S˜t≤S¯t(1)for all t≥0, a.s.
(5)
Definition 2.3.2. The set of financeable consumptionsC(S˜) is defined as a set of nonnegative, locally integrable progressively measurable processes c∈ C( ˜S) if there exist progressively measur-able processes (ϕ(0), ϕ(1), ϕ(2)) that satisfy the following two conditions:
(i) Total wealth is nonnegative.
Wt:=ϕ(0)t + ˜Stϕ(0)t +S
(2)
t ϕ
(2)
t ≥0, t≥0 (6)
(ii) The consumption stream is financeable.
Wt=W0−+ Z t 0 ϕ(1)u dS˜t+ Z t 0 ϕ(2)u dSu(2)− Z t 0 cudu, t≥0 (7)
The connection between the frictionless market problem and transaction cost problem will be described in the following two propositions.
Proposition 2.3.1. Since C( ˜S) is a set of financeable consumptions in the frictionless market, the following inequality always holds.
sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt ≥sup c∈CE Z ∞ 0 e−δtU(ct)dt (8)
Proof. To prove the Proposition 2.3.1, we prove C ∈ C( ˜S). For any c ∈ C, there exist (ϕ(0), ϕ(1), ϕ(2)) which satisfies (2). So,
ϕ(0)t +ϕ(2)t St(2)=η0+η2S0(2)+ Z t 0 ϕ(2)u dSu(2)− Z t 0 ¯ S(1)u d(ϕ(1)u )↑+ Z t 0 S(1)u d(ϕ(1)u )↓− Z t 0 cudu ≤η0+η2S0(2)+ Z t 0 ϕ(2)u dSu(2)− Z t 0 ˜ Sud(ϕ(1)u )− Z t 0 cudu
Here, inequality holds becauseS˜∈ S. Then, using intgration-by-parts, we can get
ϕ(0)t +ϕ(1)t St(1)+ϕ(2)t St(2) ≤η0+η1S˜0+η2S0(2)+ Z t 0 ϕ(1)u dSu(1)+ Z t 0 ϕ(2)u dSu(2)− Z t 0 cudu Defineϕ˜(0) as ˜ ϕ(0):=η0+η1S˜0+η2S0(2)+ Z t 0 ϕ(1)u dS(1)u + Z t 0 ϕ(2)u dSu(2)− Z t 0 cudu−ϕ(1)t S (1) t −ϕ (2) t S (2) t
Then,ϕ˜(0)≥ϕ(0) and (7) is satisfied with (ϕ˜(0), ϕ(1), ϕ(2), c). Also,
0≤ϕ(0)t +S(1)t ϕ(1)t + −S¯t(1) ϕ(1)t − +St(2)ϕ(2)t ≤ϕ˜(0)t +ϕ(1)t S˜t+ϕ(2)t S (2) t
which means that (6) is satisfied. Therefore,c∈ C( ˜S) and complete the proof.
Proposition 2.3.2. GivenS˜ ∈ S, letˆc∈ C( ˜S) solve the frictionless optimization problem, that is E Z ∞ 0 e−δtU(ˆct)dt = sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt (9)
with (ϕˆ(0),ϕˆ(1),ϕˆ(2)) that satisfies the (7). Assume that
(i) ϕˆ(1) is right-continuous process of finite variation.
(ii) (ϕˆ(0),ϕˆ(1),ϕˆ(2)) satisfies (3).
(iii) d(ϕˆ(1))↑ = 1{S˜t= ¯St}d(ϕˆ(1))↑
(iv) ˆc,ϕˆ(0),ϕˆ(1),ϕˆ(2) are continuous processes except for a possible initial jump at t = 0-.
Proof. Let (ˆc,ϕˆ(0),ϕˆ(1),ϕˆ(2)) satisfy the assumption (i) to (iv) in the proposition. Then, since it satisfies (7), we have ˆ ϕt(0)+ ˆϕ(1)t St(1)+ ˆϕ(2)t St(2) =η0+η1S˜0+η2S˜0(2)+ Z t 0 ˆ ϕ(1)u dS˜u+ Z t 0 ˆ ϕ(2)u dS˜u(2)− Z t 0 ˆ cudu
And using integration-by-part, we can get
ˆ ϕt(0)+ ˆϕ(2)t St(2)=−ϕˆ(1)t St(1)+η0+η1S˜0+η2S˜0(2)+ Z t 0 ˆ ϕ(1)u dS˜u+ Z t 0 ˆ ϕ(2)u dS˜u(2)− Z t 0 ˆ cudu =η0+η2S˜0(2)− Z t 0 ˜ Sudϕˆ(1)u + Z t 0 ˆ ϕ(2)u dS˜(2)u − Z t 0 ˆ cudu =η0+η2S˜0(2)− Z t 0 ¯ Sud( ˆϕ(1))↑u+ Z t 0 Sud( ˆϕ(1))↓u+ Z t 0 ˆ ϕ(2)u dS˜u(2)− Z t 0 ˆ cudu
which means it satisfies (2) andcˆ∈ C. And (9) and (10) imply (11).
Definition 2.3.3. Let S˜∈ S. S˜ is called a shadow price process if
sup c∈CE Z ∞ 0 e−δtU(ct)dt = sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt <∞. (11)
Remark 2.3.1. From Proposition 2.3.1. and Proposition 2.3.2. we can represent maxi-mization problem (4) to the following minimaxi-mization problem.
sup c∈C E Z ∞ 0 e−δtU(ct)dt = inf ˜ S∈S sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt (12)
III
Review of Choi [1]
In this section, we review the work of Choi. But before we review the paper, we analyze the frictionless market with two risky assets.
3.1 Frictionless market with two risky assets
Suppose that there are two risky assets with no transaction costs. We define π1, π2 by
π1 =
ϕ(1)S(1)
ϕ(0)+ϕ(1)S(1)+ϕ(2)S(2), π2=
ϕ(2)S(2)
ϕ(0)+ϕ(1)S(1)+ϕ(2)S(2)
which are the proportion of the each risky asset over risk-free and risky assets. Theorem 3.1.1. Suppose that the following condition holds.
δ > q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2
Then, the optimal policy is
c∗t =Cwt, π1∗t= (1 +q)(µ1− ρσσ21µ2) (1−ρ2)σ2 1 , π2∗t= (1 +q)(µ2− ρσ2 σ1 µ1) (1−ρ2)σ2 2 where C is defined as C = (1 +q)(δ− q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 ).
To prove Theorem 3.1.1, we first huristicaly derive the optimal investment and consumption solution. And then, we rigorously verify that these solutions are really the solutions of the objective function. This theorem and proof are given in [6] for the one risky asset case. We follow the way in [6] to prove for the two risky assets case.
3.1.1. Heuristic derivation Using Ito’s lemma, we can get
dWt=dϕ0t+S1tdϕ1t+ϕ1tdS1t+S2tdϕ2t+ϕ2tdS2t =−ctdt+ϕ1tS1t(µ1dt+σ1dB1t) +ϕ2tS2t(µ2dt+σ2dB2t) = (µ1π1tWt+µ2π2tWt−ct)dt+σ1π1tWtdB (1) t +σ2π2tWtdB (2) t
Define the function v(x) by v(x) = sup (π1,π2,c)∈A(π1,π2) E[ Z ∞ 0 e−δtU(ct)dt]
wherex=η0+η1S0(1)+η2S0(2) =W0 is the initial wealth. Assume that the processWtis finite. Then, we can use the dynamic programming principle (See Corollary 4.2 in [7]). The following lemma is suitable for the model.
Lemma 3.1.1. (Dynamic programming principle) Let (π, c)∈ A(π) be given. Then it satisfies
v(W0) = sup (π1,π2,c)∈A(π1,π2) E[ Z ∞ 0 e−δtU(ct)dt+e−δtv(Wt)] Here, π defined byW0.
Assume v is C2 function. Then, by Ito’s lemma and Lemma 3.1.1,
0 = sup (π1,π2,c)∈A(π1,π2) E[ Z ∞ 0 e−δtU(ct)dt+e−δtv(Wt)−v(W0)] = sup (π1,π2,c)∈A(π1,π2) E[ Z ∞ 0 e−δt U(c)−δv(Ws) + (µ1π1sWs+µ2π2sWs−cs)v0(Ws) + 1 2(σ 2 1π12sWs2+σ22π22sWs2+ 2σ1σ2π1sπ2sWs2ρ) ds+ Z t 0 e−δsσ1π1sWsv0(Ws)dBs(1) + Z t 0 e−δsσ2π2sWsv0(Ws)dBs(2)
Suppose that v0(x) = dxdv is bounded. Then, The last two terms are zero and Hamilton-Jacobi-Bellman(HJB) equation is made as
max(π1,π2,c)∈A(π1,π2) 1 pc p+ (µ 1π1x+µ2π2x−c)v0+ 1 2(σ 2 1π12x2+σ22π22x2+σ1σ2π1π2x2ρ)v00−δv = 0 (13)
From the FOC, the maximam is obtained at (˜c,˜π1,π˜2) where
˜ c= (v0)p−11, π˜ 1= v0 (1−ρ2)σ 1xv00 (−µ1 σ1 +µ2ρ σ2 ), π˜2= v0 (1−ρ2)σ 2xv00 (−µ2 σ2 +µ1ρ σ1 ) Then (13) changes to −δv+1−p p (v 0 ) p 1−p− 1 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 (v0)2 v00 = 0 (14) Definev˜(x)as ˜ v(x) = 1 pC p−1xp
Then, ˜v(x) satisfies (14) and maximizing π1, π2, c under ˜v(x) are equal to π˜1,π˜2,˜c. These
˜
3.1.2. Verification of the optimal strategy
In this subsection, we will see that the candidate solution we derived in section 3.1.1 is an optimal solution. Let (π1, π2, c)∈ A(π1, π2) be an arbitrary policy. Then, Wt is given by
Wt=e Rt 0(µ1π1s+µ2π2s− 1 2σ 2 1π12s−12σ 2 2π22s−σ1σ2π1sπ2sρ)ds+R0tσ1π1sdBs(1)+R0tσ1π2sdB(2)s (W0− Z t 0 cse− Rt 0(µ1π1s+µ2π2s− 1 2σ 2 1π12s−12σ 2 2π22s−σ1σ2π1sπ2sρ)ds− Rt 0σ1π1sdB (1) s − Rt 0σ1π2sdB (2) s )
From Hölder inequality and boundness of π1t, π2t, Wt has finite moments of all orders. Define
Mt as
Mt=
Z t
0
e−δsU(cs)ds+e−δtv˜(Wt) From Ito’s lemma,
Mt−M0 = Z t 0 e−δs (µ1π1sWs+µ2π2s−cs)˜v0(Ws) + 1 2(σ 2 1π21x2+σ22π22x2+σ1σ2π1π2x2ρ)˜v00(Ws) +σ1Cp−1 Z t 0 e−δsπ1sWspdB(1)s +σ2Cp−1 Z t 0 e−δsπ2sWspdB(2)s Sinceπ1s, π2s andWtp is bounded (W
p
t is bounded from Jensen’s inequality), expectation of last two terms are zero. Also, Since the integrand of the first term is nonpositive,Mtis supermartin-gale, and is a martingale when(c, π1, π2) = (˜c,π˜1,π˜2), so that
˜
v(W0) =M0≥E[Mt] =E[
Z t
0
e−δsU(cs)]ds+e−δtE[˜v(Wt)] (15) From Ito’s lemma and wealth equation, we can see that
E[e−δtv˜(Wt)] = Cp−1W0p p E[Gte Rt 0a(s)ds], whereGt=e 1 2 Rt 0p2σ21π21s+p2σ22π22sds+ Rt 0pσ1π1sdBs(1)+ Rt 0pσ2π2sdBs(2) anda(s) =p(µ 1π1s+µ2π2s−Wcss− 1
2(1−p)(π21sσ12+π22sσ22))−δ. Note that from the Novikov’s condition,Gt is a martingale since
π1, π2 is bounded. Also, when (c,π1, π2) = (˜c,π˜1,π˜2), a(s) = −C. Therefore, the last term of
(15), e−δtE[˜v(Wt)]→0 ast→ ∞
To complete the proof, we divide the cases 0 < p < 1 and p < 0. For 0 < p < 1, a(s) has upper bound (p−1)C which is less than 0. So, the last term of (15),e−δt
E[˜v(Wt)]→0 as
t→ ∞. So, ˜v=v and (˜c,π˜1,π˜2) is an optimal.
Let’s consider the casep <0. For >0, consider
˜ v(x) = 1 pC p−1(x+)p. Then, it satisfies
3.2 Result of Choi [1]
3.2.1 Heuristic derivation of the free boundary ODE
We will heuristically derive a free boundary ODE from the HJB equation for the optimization problem (4). ForS˜∈ S, we can write S˜t=St(1)eYt for an Ito-Process Y. ThenYt∈[y,y¯], where
y:=ln(1−λ) and y¯:=ln(1 + ¯λ). Assume that the dynamics of Y is given by
dYt=mtdt+s1tdBt(1)+s2tdBt(2) (17)
for some processes m, s1, s2. Then, the state price density process H in the market with S˜ and
S(2) satisfies the stochastic differential equation
dHt=−Ht
θ1(mt, s1t, s2t)dBt(1) +θ2(mt, s1t, s2t)dBt(2)
, H0= 1 (18)
whereθ1 and θ2 are
θ1(m, s1, s2) := ρ(σ2s2−µ2) (1−ρ2)σ 2 −µ2s2−(m+µ1+s1σ1+ 1 2(s 2 1+s22))σ2 (1−ρ2)σ 2(s1+σ1) θ2(m, s1, s2) := µ2 σ2 −ρθ1(m, s1, s2)
Since the frictionless market model with stock pricesS˜andS(2)is complete, the standard duality theory can be applied(c.f. [8] Karatzas)
sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt = inf z>0 sup c E Z ∞ 0 e−δtU(ct)dt +z (η0+ ˜S0η1+S0(2)η2)−E Z ∞ 0 ctHtdt = (η0+ ˜S0η1+S (2) 0 η2)p p E Z ∞ 0 e−(1+q)δtHt−qdt 1−p
Then, We can write (12) as
inf ˜ S∈S sup c∈C( ˜S) E Z ∞ 0 e−δtU(ct)dt = inf Y0 (η0+ ˜S0η1+S0(2)η2)p p |w(Y0)| 1−p where w(y) := inf m,s1,s2 sgn(p)E Z ∞ 0 e−(1+q)δtHt−qdt Y0=y (19) The HJB equation for (19) is
inf m,s1,s2
where α(m, s1, s2) := (1 +q)δ− q(1 +q) 2 (θ 2 1+θ22+ 2ρθ1θ2), β(m, s1, s2) :=q((s1+ρs2)θ1+ (ρs1+s2)θ2), γ(s1, s2) := 1 2(s 2 1+s22+ 2ρs1s2).
To incorporate the requirementYt∈[y,y¯], we sets1t=s2t= 0wheneverYtreaches the boundary
y or y¯. From (20), it is expected that the boundary condition
w00(y) =w00(¯y) =∞. (21) To handle this boundary condition, we reduce the order by changing variable. Letx =−w0(y) andg: [x,x¯]7→Rasg(x) =w(y), wherex=−w0(¯y)and x¯=−w0(y).. Then, we can write (20) as inf m,s1,s2 −α(m, s1, s2)g(x)−(m+β(m, s1, s2))x+γ(s1, s2) x g0(x)+sgn(p) = 0 x∈[x,x¯]. (22)
Then, (21) gives g0(x) = g0(¯x) = 0 and R¯x
x g0(x)
x dx = ¯y−y.¯. So, (22) produces a boundary condition and an integral constraint :
g0(x) =g0(¯x) = 0,
Z x¯
x
g0(x)
x dx= ¯y−y.¯ (23)
Note that changing variable fromw0 to g gives order reduction.
3.2.2 Main result
In section 3.1.1, we derived the free boundary problem heuristically. In this subsection, we will see the main result of [1] without proof. Proposition 3.2.1 shows the existence for the solution of the free boundary problem in the previous subsection. And Theorem 3.1.1 provide the explicit characterization of well-posedness of the problem.
Proposition 3.2.1. Assume that the model parameters satisfy one of the following conditions: (i) p≤0, (ii) 0< p <1 andδ > 2(1−qρ2)(( µ1 σ1) 2+ (µ2 σ2) 2−2ρµ1µ2 σ1σ2), and (iii) 0< p <1, δ≤ 2(1−qρ2)(( µ1 σ )2+ ( µ2 σ )2−2ρ µ1µ2 σ σ ) and c ∗ < ln(1+¯λ 1−λ),
Then, there exist constants x, x¯ and a function g∈C2[x,x¯]that satisfy following conditions:
(1) If µ1> ρµσ22σ1, then 0< x <x¯. Ifµ1 < ρµσ22σ1, then x < x¯ < 0.
(2) For x∈[x,x¯], g satisfies the differential equation
inf m,s1,s2 −α(m, s1, s2)g(x)−(m+β(m, s1, s2))x+γ(s1, s2) x g0(x) +sgn(p) = 0
where α, β, γ are given in section 3.2.1
(3) The following boundary/integral conditions are satisfied.
g0(x) =g0(¯x) = 0 and Z x¯ x g0(x) x dx=ln 1 + ¯λ 1−λ (4) The functions qg(x), qg(x)(g0(x) + 1)−(1 +q)xg0(x), q(g(x)−xg0(x)), and g0(x) + 1 are strictly positive on [x, x¯].
(5) g0(xx) >0 for x ∈(x,x¯)
Theorem 3.2.1. The following statements are equivalent. (1) The optimization problem is well-posed, that is,
sup c∈CE Z ∞ 0 e−δtU(ct)dt <∞
(2) There exists a shadow price process.
(3) The model parameters satisfy one of the following three conditions: (i) p≤0, (ii) 0< p <1 andδ > 2(1−qρ2)(( µ1 σ1) 2+ (µ2 σ2) 2−2ρµ1µ2 σ1σ2), and (iii) 0< p <1, δ≤ 2(1−qρ2)(( µ1 σ1) 2+ (µ2 σ2) 2−2ρµ1µ2 σ1σ2) and c ∗ < ln(1+¯λ 1−λ), where c∗ is a constant defined in [1] Defnition 6.9.
IV
Asymptotic ally optimal trading and consumption strategies
In this section, we investigate the optimal trading of the liquid, illiquid risky asset and consump-tion. For convenience, we assume thatλ¯= 0 and λ=λ. Also assume that
p >0, µ1 > ρµ2σ1 σ2 , δ > q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 . (24)
This means that the proportion invested in the illiquid risky asset should be positive.
First, we consider the case when there is no transaction costs. Recall that from the Section 3.1.1, the optimal proportion of liquid, illiquid risky asset and consumptioncM, π1M, πM2 in the frictionless market are given as
cM := (1 +q) δ− q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 πM1 := (1 +q)(µ1− ρσ1 σ2 µ2) (1−ρ2)σ2 1 πM2 := (1 +q)(µ2− ρσ2 σ1 µ1) (1−ρ2)σ2 2
Andx,x, g¯ (x), g(x)in Proposition 3.2.1 have the following expansions for small transaction cost
λ: x=−qπ M 1 cM + qζ cMλ 1/3+O λ2/3 ¯ x=−qπ M 1 cM − qζ cMλ 1/3+O λ2/3 g(x) =− 1 cM + q(1−ρ2)σ21ζ2 2(1 +q)(cM)2λ 2/3+O λ2/3 g(x) =− 1 cM − q(1−ρ2)σ21ζ2 2(1 +q)(cM)2λ 2/3+O λ2/3 where ζ := 3(1 +q)(π1M)2(1−π1M)2 4 + 3(1 +q)(µ2(1 +q)−ρσ1σ2)2(π1M)2 4(1−ρ2)σ2 1σ22 13
4.1 Optimal strategy
In this subsection, we will see the optimal consumption and trading strategies of liquid and illiquid asset. The proofs are in the section 5 in [1].
4.1.1. Optimal trading of the illiquid asset
Corollary 4.1.1. Under the assumption (24), minimally trading the proportion of illiquid asset within the interval [π1,π¯1] is optimal. In othere words,
π1≤ ϕˆ (1) t S (1) t ˆ ϕt(0)+ ˆϕ(1)t St(1)+ ˆϕ(2)t St(2) ≤π¯1
Corollary 4.1.1 says that the optimal trading of the illiquid asset is maintaining the proportion of investment in the illiquid asset within some interval [π1,π¯1]. That is, [π1,π¯1] is the no
transaction region. If the proportion reaches π1 or π¯1, investor minimally buys or sells the
illiquid asset to make the fraction inside the no transaction region. Andπ1,π¯1 has the following
expansions for small transaction costλ:
π1=π1M −ζλ13 +O λ2/3 ¯ π1 =π1M +ζλ 1 3 +O λ2/3
4.1.2 Optimal trading of the liquid asset
Corollary 4.1.2. Under the assumption (24), optimal proportion of investment in the liquid risky asset has the following expansions for small transaction cost λ:
πM2 −ρσ1ζ σ2 λ1/3+O λ2/3
when selling the illiquid asset,
πM2 +ρσ1ζ σ2 λ1/3+O λ2/3
when buying the illiquid asset.
Recall that according to Corollary 4.1.1, when the fraction of the illiquid asset reaches π1
or π¯1, the optimal strategy for the investor is minimally buying or selling the illiquid asset.
Corollary 4.1.2 says that when the fraction of the illiquid asset reachesπ1orπ¯1and buys or sells
the illiquid asset, the optimal proportion of the liquid asset has the above expansion for small transaction cost λ.
4.1.3 Optimal consumption rate
Corollary 4.1.3. Under the assumption (24), optimal consumption rate proportion has the following expansions for small transaction cost λ:
ˆ ct ˆ ϕt(0)+ ˆϕ(1)1 St(1)+ ˆϕ(2)t St(2) =cM −q(1−ρ 2)σ2 1ζ2 2(1 +q) λ 2 3 +O(λ), a.s.
Recall that CM is the optimal consumption rate in the frictionless market. From Corollary 4.1.3, we can see that the optimal consumption rate in the market having transaction costs always bigger than the one without transaction costs. One possible guess of this effect is that the existence of the transaction costs makes investment less attractive, and causes an increase in the consumption rate.
V
Parameter Estimation
In this section, we will apply the real data to the model we discussed. In the previous model, we assumed zero interest rates but since interest rates are not zero in the real world, we will consider the interest rates in this section.
5.1 Data description
To apply the model, we have used data related to the liquid and illiquid assets. One is KOSPI200 data (index consists of 200 big companies of the Stock Market Division) and the other is house price index (the price of the residential real estate in Korea) data which are obtained from Korean Statistical Information Service (KOSIS) and Korea Appraisal Board respectively. The data concerns the monthly time period between January 2004 to August 2019. Figure 2 shows the graph of the KOSPI200 and house price index from January 2004 to August 2019.
The parameters for the liquid and illiquid asset are calculated as follows. First, the illiquid asset parametersµ1 andσ1 is determined as
µ1 = 1 N N X i=1 (returnS(1))i−ri (25) σ1= v u u u t PN i=1 ((returnS(1))i−ri−µ1)2 N (26)
Similary, the liquid asset parameter µ2 and σ2 is determined as
µ2= 1 N N X i=1 (returnS(2))i−ri+di (27) σ2 = v u u u t PN i=1 ((returnS(2))i−ri+di−µ2)2 N (28)
whereri and di denotes interest rate and dividend respectively. Here, return ofS(j), j= 1,2 is calculated as(returnS(j))i =
Si(j)−Si(−j)1
(a) KOSPI200 (b) House price index
Figure 2: KOSPI 200 and House price index from 2004. 01 to 2019. 08 Other parameters such as δ, p, λ, ρ will be determined as follows.
To decide the discount factorδ, we will calculate it as suggested in [3]. In [3], discount facter
β is represented as β = 1/(1 + ˜r) where r˜is real interest rate. That is, r˜i = ri+πie wherer˜i is real interest rate, ri is nominal interest rate (just interest rate we mostly use), andπe is the expected inflation rate. And by the adaptive inflation expectation, expected inflation rate is based on lagged inflation rate. The relation betweenδ in this paper andβ is
e−δ =β = 1/(1 + ˜r) (29) So, we calculate discount factorδ asδ =−log(β) =−log(1/(1 + ˜r)).
For the utility function parameter p, we will just use the value in [4]. According to [4], relative risk aversion coefficient of real estate asset owners in 2012 is 3.85. It means that(1−p) = 3.85 i.e. p=−2.85in our utility function.
Transaction cost λ mostly consists of the acquisition tax, local education tax, agricultural special tax, real estate agent’s commission and legal fees. So we will only consider these. The rates of components are listed below.
(acquisition tax) + (local education tax) + (agricultural special tax)⇒1.1∼3.5%
real estate agent’s commission⇒0.4∼0.9%
The correlationρis the correlation between KOSPI200 return and house price return. Figure 3 represents the graph of kospi200 return and house price return.
Figure 3: KOSPI200 return and House price return
5.2 Results
In this subsection, using the KOSPI200 and House price index data, we will apply to the re-sults in Section 4 and see the effect of transaction costs. The procedure will be the following. First, the estimated parameters using the formula described in Section 5.1 will be shown. And then, we check the assumptions used in Section 4. Lastly, we will apply to the results in Section 4.
As described in Section 5.1, we can get the parameter estimates as follows.
µ1 = 0.0001, σ1= 0.0047 illiquid asset
µ2 = 0.0052, σ2= 0.051 liquid asset
p=−2.85, ρ= 0.008, δ = 0.0004, λ= 0.03
Here,µ1, µ2, σ1, σ2 was calculatied by (25) to (28).
So, what we want to do is applying these estimated parameters to the results in Section 4. But before we apply the estimated parameters to the results in Section 4, we should check the assumptions in Section 4.
The assumptions in Section 4 were p >0, µ1 > ρµ2σ1 σ2 , δ > q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 .
From our estimated parameters, we can check that
p <0, µ1 > ρµ2σ1 σ2 , δ > q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 .
The only difference with the assumptions isp <0. This induces some changes in some parts of the results. For the case of transaction costλ= 0, there are no changes. So, we can calculate
CM, πM1 , π2M as cM := (1 +q) δ− q 2(1−ρ2) µ1 σ1 2 + µ2 σ2 2 −2ρµ1µ2 σ1σ2 = 0.00114 πM1 := (1 +q)(µ1− ρσ1 σ2 µ2) (1−ρ2)σ2 1 = 1.13 πM2 := (1 +q)(µ2− ρσ2 σ1 µ1) (1−ρ2)σ2 2 = 0.52 And x,x, g¯ (x), g(¯x) changes to x=−qπ M 1 cM + qζ cMλ 1/3+O λ2/3 = 331.5 ¯ x=−qπ M 1 cM − qζ cMλ 1/3+O λ2/3 = 1132.4 g(x) =− 1 cM + q(1−ρ2)σ12ζ2 2(1 +q)(cM)2λ 2/3+O λ2/3 =−883.6 g(x) =− 1 cM − q(1−ρ2)σ12ζ2 2(1 +q)(cM)2λ 2/3+O λ2/3 =−865.2 where ζ := 3(1 +q)(πM1 )2(1−π1M)2 4 + 3(1 +q)(µ2(1 +q)−ρσ1σ2)2(π1M)2 4(1−ρ2)σ2 1σ22 13 = 1.99
5.2.1 Parameter estimation in the optimal trading of the illiquid asset.
According to optimal strategy of illiquid asset in Section 4, trading the proportion of illiquid asset within the interval[π1,π¯1]is optimal. In this case,p <0doesn’t generate any change. So,
π1,π¯1 can be calculated as π1 =πM1 −ζλ13 +O λ2/3 = 0.51 ¯ π1=π1M+ζλ 1 3 +O λ2/3 = 1.75
where transaction cost λ = 0.03. In other words, the optimal trading of the illiquid asset is maintaining the proportion of investment in the illiquid asset as
π1= 0.51≤ ϕˆ (1) t S (1) t ˆ ϕt(0)+ ˆϕ(1)t St(1)+ ˆϕ(2)t St(2) ≤1.75 = ¯π1
Figure 4 shows the optimal trading of the illiquid asset with different transaction costs and Table 1 represents the values of π1,π¯1 in terms of percentage. For example, suppose that the
transaction cost is 1%. Then, it is optimal not to trade illiquid asset when the proportion of investment in the illiquid asset is within70%to 156%. From Figure 4 and Table 1, We can see that transaction cost does have an effect on the interval [π1,π¯1]. It is because the coefficient of
λ13, which represents the sensitivity of the increase in the investment proportion of the illiquid
risky asset due to transaction costs, is about 1.99 which is large compared to the liquid asset.
λ
(%)
0
1
2
3
4
¯
π
1(%)
113.082 155.975 167.124 174.945 181.172
π
1(%)
113.082
70.189
59.040
51.219
44.993
Table 1: Values of π1,π¯1 on certain points
5.2.2 Parameter estimation in the optimal trading of the liquid asset.
In Section 4, we have seen the optimal proportion of the liquid asset when investor sells or buys the illiquid asset. Similar to the case of illiquid asset,p <0 doesn’t generate any change in the case of liquid asset. So,π2,π¯2 can be calculated as
π2 :=πM2 −ρσ1ζ σ2 λ1/3+O λ2/3
= 0.518 when selling the illiquid asset
¯ π2 :=π2M + ρσ1ζ σ2 λ1/3+O λ2/3
= 0.519 when buying the illiquid asset
where transaction cost λ= 0.03. In other words, when the invested proportion of the illiquid asset reaches π¯1 = 1.75, investor sells illiquid asset and the proportion of the liquid asset is
π2 = 0.518 at that point. Similarly, when the invested proportion of the illiquid asset reaches
π1 = 0.51, investor sells illiquid asset and the proportion of the liquid asset isπ¯2 = 0.519 at that
point.
Similar to illiquid risky asset case, Figure 5 shows the optimal trading of the liquid asset with different transaction costs and Table 2 represents the values of π2,π¯2 in terms of percentage.
For example, suppose that transaction cost is 1%. Then, when an investor sells (respectively buys) illiquid asset, the optimal proportion of the liquid asset is51.876%(respectively51.813%). From Figure 5 and Table 2, We can see that transaction cost have less effect on the interval [π2,π¯2]. It is because the coefficient of λ
1
3, which represents the sensitivity of the increase in
the investment proportion of the liquid risky asset due to transaction costs, is 0.00147 which is smaller than the case of the illiquid risky asset.
Figure 5: Proportion of investment in the liquid asset
λ
(%)
0
1
2
3
4
¯
π
2(%)
51.844 51.876 51.885 51.890 51.895
π
2(%)
51.844 51.813 51.805 51.799 51.795
Table 2: Values of π2,π¯2 on certain points
5.2.3 Parameter estimation in the optimal consumption rate
In Section 4, we got the asymptotic expansion of the optimal consumption rate proportion for small transaction cost λ. In this case,p <0 changes the result of the before. The only change is the sign in the coefficient ofλ23. So, the optimal consumption rate proportion is
ˆ ct ˆ ϕt(0)+ ˆϕ(1)1 St(1)+ ˆϕ(2)t St(2) =cM −q(1−ρ 2)σ2 1ζ2 2(1 +q) λ 2 3 +O(λ) = 0.00116, a.s.
Figure 6 shows the optimal consumption rate with different transaction costs and Table 3 represents the values of it in terms of percentage. For example, when transaction cost is 1%, the optimal consumption rate is 0.11494%. Similar to the case of a liquid asset, the optimal consumption rate isn’t much affected by the transaction cost λ. It is because the coefficient of
λ23, the sensitivity of the increase in the consumption rate due to transaction costs is 0.000125
which is small compared to the illiquid asset case. Also, we can see that when transaction cost changes0%to4%, optimal consumption rate changes to0.1144%to0.1158%. That is, compared to the case having no transaction costs, the investor increases his consumption rate by about 1.2%.
Figure 6: Proportion of investment in the illiquid asset
λ
(%)
0
1
2
3
4
c(%)
0.11436 0.11494 0.11528 0.11556 0.11582
Table 3: Values of optimal consumption rate on certain pointsWe can have the following interpretation. First, Note that when the transaction cost λ= 0, the optimal consumption rate is CM. So, since the coefficient ofλ23, −q(1−ρ
2)σ2 1ζ2
2(1+q) = 0.000125
which is positive, we can see that the optimal consumption rate proportion is always greater than CM in the market having transaction costs. One of the possible explanation is that the existence of the transaction costs makes the investment unappealing. So the investor increases his consumption rate.
Also, recall that −q(1−ρ2)σ12ζ2
2(1+q) , the coefficient of λ
2
3 represents the sensitivity of the increase
in the consumption rate which is generated by the transaction costs. From [1], the original market model is equal to the market having only illiquid asset when µ2 = ρ = 0. Therefore,
to compare with the effect of the transaction costs on consumption where the market model has the illiquid asset only, we consider
− q(1−ρ2)σ12ζ2 2(1+q) µ2=ρ=0
, that is, the sensitivity of the increase in the consumption rate which is generated by the transaction costs when there is only an illiquid asset.
From the data, we can get
− q(1−ρ2)σ21ζ2 2(1+q) µ2=ρ=0
= 0.00000129. So, we can get the following inequality. −q(1−ρ 2)σ2 1ζ2 2(1 +q) ≥ −q(1−ρ 2)σ2 1ζ2 2(1 +q) µ2=ρ=0 , if ρ= 0
It means that the coefficient of λ23 in the market having liquid and illiquid assets is larger
than the market having illiquid assets only. In other words, The effect of the transaction costs on consumption is more noticeable when there are liquid and illiquid assets. Since ρ = 0, the optimal investment proportion of the illiquid risky asset is π1M in both markets. The existence of the liquid asset makes investor also take an exposure to the risk of the liquid risky asset, That isπ2M 6= 0. And this induces the increase in the volatility of the total wealth process. Therefore, the market having liquid asset trading opportunity has more frequent trading of the illiquid asset (from the optimal trading of the illiquid asset) and the model has more trading costs due to rebalancing. So, in the market with a liquid asset, the effect of the transaction costs on the consumption is stronger.
Figure 7: consumption rate with illiquid asset only
λ
(%)
0
1
2
3
4
c(%)
0.014742 0.014748 0.014751 0.014754 0.014757
Table 4: Values of optimal consumption rate on certain points with illiquid asset onlyFigure 7 and Table 4 shows the effect of the transaction cost to the consumption rate with the illiquid asset only. For example, suppose that the transaction cost is1%. Then, the optimal consumption rate is 0.014748%when there is only an illiquid asset. And when transaction cost changes 0% to 4%, the optimal consumption rate changes to0.01474% to 0.01476%. That is, compared to the case having no transaction costs, the investor increases his consumption rate by about0.14%. Also, we can see that the incremental amount of the optimal consumption rate is much smaller than the case with a liquid asset. In other words, compared with the market having a liquid asset, the optimal consumption rate is less affected by the transaction cost λ.
VI
Conclusion
In this paper, we have considered the model with two risky assets, which are liquid and illiquid assets. From [1], we could get the asymptotic ally optimal trading and consumption strategies as in Section 4. The optimal investment of the illiquid asset is trading the proportion of illiquid asset within some interval(no transaction region). And using these strategies, we applied to the real data. For the liquid asset, we considered monthly KOSPI200 data which is the index consists of 200 big companies of the Stock Market Division in Korea. For the illiquid asset, we used monthly house price index data which represents the price of the residential real estate in Korea. Using these two data, we obtained the optimal investment and consumption rate and checked how they change as transaction costs become different. Also, we calculated the optimal consumption rate in the market having illiquid asset only to compare it with the optimal consumption rate in the original model.
The optimal strategies we obtained are the following. First, the optimal investment of illiquid asset is investing 113%of the total wealth to the illiquid asset and investor doesn’t trade when it is within 51% to 175% of the total wealth(no transaction region). And for the liquid asset, the optimal investment of the liquid asset is investing 51.844%of the total wealth to the liquid asset. And when an investor sells(respectively buys) the illiquid asset, the optimal proportion of the liquid asset is51.8%(respectively, 51.9%). Lastly, the optimal strategy of the consumption rate is consuming0.12% of the total wealth in a month.
For example, suppose that we have the total wealth 1,000M KRW. Then, we have the following optimal strategies. First, we lend money from the bank about 650M KRW. And then, we invest around 1,130M KRW to the residential real estate and don’t sell or buy when it is from 510M KRW to 1,750M KRW. Also, we invest 518.4M won to the stock and when selling(respectively buying) the illiquid asset, the amount of the investment in the stock is 518M KRW(respectively, 519M KRW) And optimal consumption rate is consuming 120M KRW in a month.
As transaction costs change, we could see that the effect of the transaction costs is large on the illiquid asset. But illiquid assets and consumption are less affected by the transaction costs. Compared with the model having an illiquid asset only, the effect of the transaction costs on the consumption is more noticeable. It is because the existence of the liquid asset causes an increase in the volatility of the total wealth process, and therefore more frequent trading of the illiquid asset.
We have seen that the existence of the liquid asset does have an effect compared to the market having illiquid asset only. So we can conclude that It is important to consider liquid asset when analyzing the market having an illiquid asset.
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